দেওয়া আছে, $$ \begin{aligned} \cos \theta&=\frac{1}{2}\left(a+\frac{1}{a}\right) \end{aligned} $$ আমরা জানি, $$ \begin{aligned} \cos 3\theta&=4\cos^3 \theta - 3\cos \theta\\ &=4\left[\frac{1}{2}\left(a+\frac{1}{a}\right)\right]^3 - 3\left[\frac{1}{2}\left(a+\frac{1}{a}\right)\right]\\ &=4 \times \frac{1}{8}\left(a+\frac{1}{a}\right)^3 - \frac{3}{2}\left(a+\frac{1}{a}\right)\\ &=\frac{1}{2}\left[\left(a+\frac{1}{a}\right)^3 - 3\left(a+\frac{1}{a}\right)\right]\\ &=\frac{1}{2}\left(a^3+\frac{1}{a^3}+3a \cdot \frac{1}{a}\left(a+\frac{1}{a}\right) - 3\left(a+\frac{1}{a}\right)\right)\\ &=\frac{1}{2}\left(a^3+\frac{1}{a^3}\right) \end{aligned} $$